Multi-pulse homoclinic phenomena in resonant Hamiltonian systems

Citation

Haller, Gyorgy (1994) Multi-pulse homoclinic phenomena in resonant Hamiltonian systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/dx2n-8q82. https://resolver.caltech.edu/CaltechETD:etd-11302007-152315

Abstract

In this thesis we develop a global perturbation method to detect homoclinic orbits which arise in perturbations of manifolds of equilibria with a homoclinic structure in two degree-of-freedom Hamiltonian systems. Our energy-phase method gives conditions for the existence of multiple-pulse and "jumping" orbits asymptotic to different invariant sets within a slow manifold of the perturbed system. The perturbations we consider are either Hamiltonian or weakly dissipative, and the orbits created by these perturbations are generic in both cases. The geometric criterion we derive requires simple algebraic manipulations and detects orbits which are not amenable to Melnikov-type methods, even if those methods are combined with geometric singular perturbation theory. We apply the energy-phase method to the analysis of three-degree-of-freedom resonant Hamiltonian normal forms and prove the existence of non-exponentially small splittings of separatrices connecting invariant tori. These structures, together with double-pulse homoclinic tori, exist arbitrarily close to resonant elliptic equilibria in a class of Hamiltonian systems. As another application, we consider a two-mode truncation of the driven nonlinear Schrodinger equation and establish the existence of chaotic multiple-pulse "jumping orbits" which can be arranged in a fractal structure. We confirm the predictions of the energy-phase method by numerical simulation and visualization of intersecting multipulse orbit cylinders.

Thesis Files